Abstract
We provide a necessary and sufficient condition for strict local minimisers of differences of convex (DC) functions, as well as related results pertaining to characterisation of (non-strict) local minimisers, and uniqueness of global minimisers.
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Attouch H., Wets R.J.B.: Quantitative stability of variational systems: II. A framework for nonlinear conditioning. SIAM J. Optim. 3(2), 359–381 (1993)
Dür M.: A parametric characterization of local optimality. Math. Methods Oper. Res. 57, 101–109 (2003)
Hiriart-Urruty, J.B.: Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. In: Convexity and Duality in Optimization: Proceedings of the Symposium on Convexity and Duality in Optimization Held at the University of Groningen, The Netherlands, no. 256 in Lecture notes in Economics and Mathematical Systems, pp. 37–70. Springer, New York 22 June 1984
Hiriart-Urruty J.B.: From convex optimization to non convex optimization, Part I: necessary and sufficent conditions for global optimality. In: Clarke, F., Demyanov, V., Giannessi, F. (eds) Nonsmooth Optimization and Related Topics, pp. 219–239. Plenum Press, New York (1988)
Hiriart-Urruty J.B., Lemaréchal C.: Convex Analysis and Minimization Algorithms I–II. Springer, New York (1993)
Martínez-Legaz J.E., Seeger A.: A formula on the approximate subdifferential of the difference of convex functions. Bull. Aust. Math. Soc. 45(1), 37–41 (1992)
Penot J.P.: On the minimization of difference functions. J. Global Optim. 12, 373–382 (1998)
Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1972)
Valkonen, T.: Diff-convex combinations of Euclidean distances: a search for optima. No. 99 in Jyväskylä studies in computing. Ph.D Thesis, University of Jyväskylä (2008)
Valkonen, T., Kärkkäinen, T.: Clustering and the perturbed spatial median. Submitted (2008)
Valkonen, T., Kärkkäinen, T.: Continuous reformulations and heuristics for the Euclidean travelling salesperson problem. ESAIM Control Optim. Calc. Var. 15(4), (2009). doi:10.1051/cocv:2008056
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Valkonen, T. Refined optimality conditions for differences of convex functions. J Glob Optim 48, 311–321 (2010). https://doi.org/10.1007/s10898-009-9495-y
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DOI: https://doi.org/10.1007/s10898-009-9495-y